体育馆(tǐ yù guǎn) - sports centre, gymnasium in Mandarin

体育馆 (tǐyùguǎn) is a sports centre or gymnasium, an indoor building for sport. 体育 (tǐyù) means "physical education, sport" and 馆 (guǎn) means "hall, building", the same 馆 found in 图书馆 (túshūguǎn), "library", and 博物馆 (bówùguǎn), "museum". Whenever you see 馆 on the end of a two-syllable word, it usually marks a public building of some kind.

It is where 运动 (yùndòng), "sport, exercise", actually happens indoors: basketball, badminton and swimming pools are all commonly housed in a 体育馆. A large stadium built mainly for outdoor sport is usually 体育场 (tǐyùchǎng) instead, with 场 (chǎng), "open ground", rather than 馆.

学校体育馆 (xuéxiào tǐyùguǎn), "school sports centre", and 去体育馆锻炼 (qù tǐyùguǎn duànliàn), "go to the sports centre to exercise", are the two most natural collocations to learn alongside it.